Positive Matrix Factorization (Pmf)

SOURCE ALLOCATION AND VISIBILITY IMPAIRMENT

IN TWO CLASS I AREAS WITH POSITIVE MATRIX FACTORIZATION

Keith A. Rose

Senior Environmental Scientist

U.S. Environmental Protection Agency, Region 10

June 13, 2005

Introduction

In the 1977 amendments to the Clean Air Act (CAA), Congress set a national goal of improving visibility in mandatory Class I areas by controlling sources of visibility- impairing pollutants. In 1988, the States, Federal Land Managers, and Environmental Protection Agency (EPA) initiated the IMPROVE monitoring program to measure speciated fine particulate (PM2.5) concentrations in national parks and wilderness areas. The purpose of this monitoring program was to identify which pollutants are causing impairment of visibility in Class I areas, and to identify the sources responsible for these pollutants. In 1999, in compliance with the CAA, EPA issued regulations requiring States to develop implementation plans to control sources that contribute to visibility impairment in all 156 mandatory Class I national parks and wilderness areas.

In this study, Positive Matrix Factorization (PMF) was used to analyze IMPROVE monitoring data collected at two West Coast Class I areas over two time periods, 1991-95 and 2000-03. These Class I areas were Mt. Rainier National Park in Washington, and Yosemite National Park in California. PMF identified source profiles associated with each air pollution source and the time-dependent contribution of each source to fine particulate concentrations in these two Class I areas. These source profiles and time-dependent concentrations were used to determine the light extinction (a measurement of visibility impairment) caused by each source at Mt. Rainier and Yosemite National Parks. The results show that PMF can be used as a tool to help determine which sources have the most significant impact on visibility in Class I areas, and how the visibility impairment from each source varies over time.

Methods

PMF Model

PMF is a variant of Factor Analysis with non-negative factor elements. It is a factor analysis method with individual weighting of matrix elements first described by Paatero and Tapper, and Paatero (1997). The PMF approach can be used to analyze 2-dimensional and 3-dimensional matrices. The 2-demensional version of PMF was used to analyze the Class I area data in this study. PMF solves the equation:

X = GF + E

In this equation, “X” is the matrix of measured values, “G” and “F” are the factor matrices to be determined, and “E” is the matrix of residuals, the unexplained part of “X”. In the PMF model, the solution is a weighted Least Squares fit, where the known standard deviations for each value of “X” are used for determining the weights of the residuals in matrix “E”. The objective of PMF is to minimize the sum of the weighted residuals. PMF uses information from all samples by weighting the squares of the residuals with the reciprocals of the squares of the standard deviations of the data values.

In environmental pollution problems, one row of “X” would consist of the concentrations of all chemical species in one sample, and one column of “X” would be the concentration of one species for each of the samples. One row of the computed “F” matrix would be the source profile for one source, and the corresponding column of “G” would be the amount of this source in each individual sample. Required input matrices for PMF are “X”, the measured values, and “Xstd-dev”, the standard deviations (uncertainties) of the measured values. PMF requires that all values and uncertainties are positive values, therefore missing data and zero values must be omitted or replaced with appropriate substitute values.

Model Operating Parameters

For analysis of the IMPROVE data, PMF was run in the robust mode suggested for analyzing environmental data by Paatero (1996). In the robust mode, the standard deviations used for weighting the residuals are dynamically readjusted through an iterative process. This process prevents excessively large values in the data set from disproportionally affecting the results.

PMF provides error models to calculate the standard deviations of the data values. According to Paatero (1996), recommended error models for environmental data include the lognormal distribution model and the heuristically-computed model. The lognormal model works well if the data have a lognormal distribution, but that is not always the case for environmental data. In the PMF analysis of particulate data collected in Hong Kong, Lee et. al. achieved good results with the heuristically-computed error model. In this study the heuristically-computed model was chosen for analysis of IMPROVE data.

Adjustment of PMF Source Concentrations

In this study, the daily PMF calculated concentrations for each source (G matrix) were adjusted through a linear regression with the measured total concentrations. The linear regression was accomplished by using the “LINEST” function in Excel. This function provides three parameters that indicate the “goodness of fit” between the measured concentration and the sum of the calculated concentrations. These parameters are “r2”, the slope of the regression line, and the uncertainty in each source regression adjustment factor. The best fit is achieved when the regression parameters “r2” and “slope” each equal 1.0, and the uncertainty in each regression factor is smaller than the value of the corresponding regression factor.

Determining the Number Sources

The most difficult challenge in using PMF to evaluate environmental data is determining the number of sources that are contributing to the contaminants collected at the monitor. In this study, five, six and seven-source solutions were generated for both Class I areas. A two-step process was used to determine which solutions generated by PMF provided the most feasible number of sources for each Class I area. First, the generated source profiles were compared to source profiles identified in previous published PMF studies. Specifically, the source profiles for each solution (F matrices) were compared to the Columbia Gorge PMF source profiles (Rose) and to those identified by the PMF analysis of Seattle IMPROVE data (Maykut et. al.). Second, the “goodness of fit” for each solution was examined to see which solutions had the best linear regression between measured and calculated source concentrations. The results of this two-step process, to identify source profiles and determine the “goodness of fit”, are shown in Table 1.

Table 1. Evaluation of PMF Solutions for Mt. Rainier and Yosemite National Parks

These results show that the six-source solutions for Mt. Rainier for 1991-95 and 2000-03 generated acceptable results. The seven-source solution for Mt. Rainier for 1991-95 was also acceptable, while the seven-source solution for 2000-03 was not. For Yosemite, the six and seven-source solutions for 1991-95 were acceptable, while only the five-source solution was acceptable for 2000-03. In all cases where there were acceptable results from multiple solutions, the solution with the higher number of sources always contained a diesel-powered mobile source and a gasoline-powered mobile source profile. In order to directly compare the total mobile (combined diesel and gasoline) source contributions between the two time periods, only those solutions which contained a combined mobile source profile were used for further analysis in this study.

Data Selection

Data used for each Class I area in this analysis were from the years 1991-95 and 2000-03. Dates that had missing data, and species that had substantial values below the laboratory minimum detection limit (MDL), were eliminated from this analysis. Species used in this analysis included: calcium, copper, elemental carbon fractions (EC1 and EC2), iron, potassium, hydrogen, sodium, lead, organic carbon fractions (OC2, OC3 and OC4), nitrate, sulfate, sulfur, silicon, and zinc. Data and data uncertainties reported as “zero” by the laboratory were replaced with a value of ½ the MDL.

Results

Identification of Source Profiles

PMF source profiles for each Class I Area for are shown in Appendix A. PMF generated four source profiles that had relatively small amounts of organic or elemental carbon and contained significant amounts one or more inorganic species. These source profiles were similar to non-combustion source profiles generated by PMF analysis of Columbia Gorge IMPROVE data. The inorganic species in each profile and its associated source are shown in Table 2.

Table 2. PMF Inorganic Profiles and Associated Sources

For each Class I area, PMF also generated two combustion source profiles similar to combustion source profiles generated by PMF for Columbia Gorge and Seattle. The organic and inorganic composition of each combustion profile and its corresponding source are shown in Table 3.

Table 3. PMF Combustion Profiles and Associated Sources

The biomass burning profiles contain the highest amounts of organic carbon, a large OC3 fraction, relatively smaller amounts of EC1 and EC2, and potassium. Mobile (combined gasoline and diesel) source profiles contain the highest amounts of EC, moderate amounts of OC, and trace amounts of iron, lead and zinc.

Source PM2.5 Concentrations

The average and 90 percentile daily PM2.5 concentrations from each source in each Class I area, for both the 1991-95 and 2000-03 time periods, are shown in Tables 4 and 5. Biomass burning contributed the largest amount of fine particulates in both Class I areas for both time periods. However, between the two time periods, average biomass burning concentrations at Mt. Rainier decreased by 42%, while biomass burning concentrations at Yosemite increased 27%. Secondary sulfate contributed the second highest amount of fine particulates at both Class I areas. Between 1991-95 and 2000-03, average concentrations of the secondary sulfate source decreased by 37% at Mt. Rainier, and by 23% at Yosemite.

Table 4. PM2.5 Concentration by Source at Mt. Rainier (ug/m3)

Table 5. PM2.5 Concentration by Source at Yosemite (ug/m3)

* The Yosemite 2000-03 solution did not include a marine source.

Time-Dependent Source Concentrations

Trends in source concentrations for both Class I areas for the 1991-95 time period are shown in Appendix B. At Mt. Rainier, secondary sulfate, secondary nitrate, soil and marine sources showed seasonal trends. At Yosemite, biomass, secondary sulfate, secondary nitrate, soil and marine sources showed seasonal trends. Months of the year during which each source made its highest contribution at each site are shown in Table 6.

Table 6. Months of Highest Source Contribution

Visibility Impairment Due to Source Emissions

Visibility impairment caused by fine particles, expressed in terms of the light extinction coefficient Bext (units of inverse megameters, 1/Mm), is given in equation 3.8 in Chapter 3 of the report titled “Spatial and Seasonal Patterns and Temporal Variability of Haze and its Constituents in the United States: Report III” (Malm et. al.):

Bext = (3 m2/g) Ft(RH)[sulfate] + (3 m2/g)Ft(RH)[nitrate] + (4 m2/g)[OC] +

(10 m2/g)[EC] + (1 m2/g)[soil]

Where: Ft(RH) = annual average relative humidity factor

The Bext for the secondary sulfate and secondary nitrate sources were determined by assuming that these sources consisted only of ammonium sulfate and ammonium nitrate, respectively. The Bext for the biomass and mobile sources were determined by assuming that the only visibility impairing components in these sources were OC and EC. Ft(RH) for Mt. Rainier was set at a value of 4.5, and the Ft(RH) for Yosemite was set at a value of 2.1. Using this approach, the average and 90 percentile Bext due to each source, based on average and 90 percentile concentrations of each source (tables 4 and 5), are shown in Tables 7 and 8.

Table 7. Average Source Bext (1/Mm)

Mt. Rainier

Yosemite

Yosemite

Table 8. 90 Percentile Source Bext (1/Mm)

Mt. Rainier

Yosemite

Yosemite

Figures 1 through 4 show the average percent of visibility impairment due to each source, relative to the total visibility impairment for all sources, in each Class I area for the 1991-95 and 2000-03 time periods.

Figure 1. Percent Source Visibility Impairment at Mt. Rainier for 1991-95

Figure 2. Percent Source Visibility Impairment at Mt. Rainier for 2000-03

Figure 3. Percent Source Visibility Impairment at Yosemite for 1991-95

Figure 4. Percent Source Visibility Impairment at Yosemite for 2000-03

Tables 7 and 8 show that that the most significant source of visibility impairment at Mt. Rainier is 1 and 2 show that the relative percents of visibility impairment from secondary sulfate, and the second most significant source at Mt. Rainier is biomass burning. Figures all sources at Mt. Rainier remained about the same for 1991-95 and 2000-03. Tables 7 and 8 also show that the most significant source of visibility impairment at Yosemite was biomass burning, and the second largest source was secondary sulfate. Figures 3 and 4 show that the relative percents of visibility impairment due to biomass burning substantially increased at Yosemite between 1991-95 and 2000-03, while the relative percents due to secondary sulfate substantially decreased between these two time periods. Visibility impairment due to secondary nitrate was the third largest of all sources at both Class I areas, and visibility impairment due to mobile sources was the fourth largest.

Conclusions

PMF generate source profiles for biomass burning, secondary sulfate, secondary nitrate, mobile sources, soil and marine aerosols that contribute to fine particulate concentrations measured at Mt. Rainier and Yosemite National Parks. The trends in these source concentrations were also identified. At Mt. Rainier, secondary sulfate, secondary nitrate, soil and marine aerosols showed seasonal trends. At Yosemite, biomass, secondary sulfate, secondary nitrate, soil and marine aerosols showed seasonal trends.

Biomass burning was responsible for the highest average concentrations of fine particulates in both Class I areas for the 1991-95 and 2000-03 time periods, and the second highest concentrations were due to secondary sulfate. At Mt. Rainier, average concentrations of particulates due to biomass burning and secondary sulfate decreased between 1991-95 and 2000-03. At Yosemite, average concentrations due to biomass burning increased between these two time periods, while concentrations of secondary sulfate decreased.

Average and 90 percentile source concentrations were used to determine the average and 90 percentile visibility impairment due to each source for both the 1991-95 and 2000-03 time periods. At Mt. Rainier, the largest source of visibility impairment was secondary sulfate, and the second largest source was biomass burning. At Yosemite, the largest source of visibility impairment was biomass burning, and the second largest source was secondary sulfate. At Mt. Rainier, average visibility impairment due to secondary sulfate decreased from a Bext value of 15.9 1/Mm in 1991-95 to 10.0 1/Mm in 2000-03, and average visibility impairment due to biomass burning decreased from 12.7 1/Mm to 7.4 1/Mm between these two time periods. At Yosemite, average visibility impairment due to biomass burning increased from a Bext value of 8.5 1/Mm in 1991-95 to 10.8 1/Mm in 2000-03, and average visibility impairment due to secondary sulfate decreased from 6.9 1/Mm to 3.9 1/Mm between these time periods. Visibility impairment due to secondary nitrate was the third largest of all sources at both Class I areas, and visibility impairment due to mobile sources was the fourth largest.

References

Chow, J., and Watson, J., Western Washington 1996-97 PM2.5 Source Apportionment Study, 1998.

Kim et. al., Factor Analysis of Seattle Fine Particles, submitted to AS&T for publication in Oct. 2003.

Lee et al., Application of Positive Matrix Factorization in Source Apportionment of Particle Pollutants in Hong Kong, Atmospheric Environment, 33, 3201-3212.

Malm et. al., Spatial and Seasonal Patterns and Temporal Variability of Haze and its Constituents in the United States: Report III, CIRA, ISSN: 0737-5352-47, May 2000.

Maykut et. al., Source Apportionment of PM2.5 at an Urban IMPROVE Site in Seattle Washington, Environmental Science and Technology, 2003, 37, 5135-5142.

Paatero, P., and U. Tapper, Positive matrix factorization: a non-negative factor model with optimal utilization of error estimates of data values, Environmetrics, 5, 111-126, 1994.

Paatero, P., User’s Guide for Positive Matrix Factorization Programs PMF2.EXE and PMF3.EXE, University of Helsinki, Helsinki, 1996.

Paatero, P., Least squares formulation of robust non-negative factor analysis, Chemometrics and Intelligent Laboratory Systems, 37, 23-35, 1997.

Rose, K., Source Allocation of Columbia Gorge IMPROVE Data with Positive Matrix Factorization, Appendix E of “Chemical Concentration Balance Source Apportionment of PM2.5 Aerosol in the Columbia River Gorge”, Oregon Department of Environmental Quality, March 31, 2003.